Proof.   1. Let \(U=|\psi _a|\)  and \(p\in \mathcal {Z}_3(U)\) .
If \(p\in D\)  then \(U(p)=0\)  and \(\nabla U(p)=(0,0)\) 
([1]}). It follows that \(p\)  is a
critical point for \(\psi _a\)  at level 0. By standard
theory of harmonic functions the zero set of \(\psi _a\)  around
a critical point at level 0 is made by (at least) 4
half-lines, meeting with equal angles.
We infer that locally around \(p\)  the function \(\psi _a\) 
defines \(q\)  nodal components with \(q\ge 4\)  and \(q\) 
is even because \(\psi _a\)  has alternate positive or
negative sign on adjacent sets.
